252solngr1 9/16/05 (Open this document in 'Page Layout' view!)
Graded Assignment 1
Please show your work! Neatness and whether the papers are stapled may affect your grade.
1. A Psychiatrist is treating a group of aborigines who are suffering from depression. Whether justifiably or not, she considers this group a random sample of 15 taken from a very large number of depressed individuals. The numbers below represent the measurement of the sample’s level of depression an hour after taking the pill using a commonly used (Coolidge Axis II) scale for measuring depression. Personalize the data as follows: add the digits of your student number to the last six numbers. Example: Ima Badrisk has the student number 123456; so the last six numbers become {51, 52, 53, 54, 55, 56}.
52 53 58 50 53 58 55 66 53 50 50 50 50 50 50
1. Compute the sample standard deviation using the computational formula. Use this sample standard deviation to compute a 99% confidence interval for the mean. The doctor believes that subjects fed a sugar pill will have an average score on the same scale of 58.73. Does the mean from your sample differ significantly from 58.73? Why?
2. How would these results change if these individuals were a random sample of 15 taken from the 150 members of the tribe that are depressed?
3. Assume that the population standard deviation is 4.50 (and that the sample of 15 is taken from a very large population). Find using the Normal table (If you have several values of that you can use, pick the average of the extreme ones.) and use it to compute a 99.5% confidence interval. Does the mean differ significantly from 58.73 now? Why?
Solution: There are two basic observations. 1) You can’t answer a question you haven’t read. It says ‘computational formula’ in the first part. If you don’t know what that means, find out! 2) You can’t do an assignment based on problems if you haven’t looked at the problems. The first 3 problems were based on Problems A1, A2 and 8.20. If you had made these your own, there was no chance of error.
1)
252solngr1 9/16/05 (Open this document in 'Page Layout' view!)
index
1 52 2704
2 53 2809
3 58 3364
4 50 2500
5 53 2809
6 58 3364
7 55 3025
8 66 4356
9 53 2809
10 51 2601
11 52 2704
12 53 2809
13 54 2916
14 55 3025
15 56 3136
819 44931
, ,
The formula for the sample standard deviation is in Table 20 of the Supplement.
.
252solngr1 9/16/05 (Open this document in 'Page Layout' view!)
From Table 3 is the formula for a two sided confidence interval when the population standard deviation is unknown. or 51.598 to 57.602.
If we ask if the mean is significantly different from 58.73, our null hypothesis is and since 58.73 is not between the top and the bottom of the confidence interval, reject and say that the mean is significantly different from 58.73. (But it is not significantly different from 57!)
2) If , the sample of 15 is more than 5% of the population, so use .
Recall that , , , and that is the formula for a two sided interval. or 51.020 to 58.180. The interval is smaller, but it doesn’t change anything – the mean is still significantly different from 58.73 (but not 58).
3) a) Find and compute a 99.5% confidence interval for the population mean.
Make a diagram! The diagram for will be a Normal curve centered at zero and will show one point, , which has 0.25% above it (and 99.75% below it!) and is above zero because zero has 50% below it. Since zero has 50% above it, the diagram will show 49.75% between zero and .
From the diagram, we want one point so that or . On the interior of the Normal table we can find to .4975 exactly. In fact, it says for 2.81. This means that we will say .
Check: This is verified graphically below.
b) We know that , and . So
=1.1629. The 99.5% confidence interval has or , so . The confidence interval is or 51.33 to 57.87. If we test the null hypothesis against the alternative hypothesis , since 58.73 is not on the confidence interval, we reject the null hypothesis or say that our results do not indicate that the mrean is significantly different from 58.73.
Check of results in 1 and 3 using Minitab.
————— 9/16/2006 3:19:47 AM ————————————————————
Welcome to Minitab, press F1 for help.
MTB > WOpen "C:\Documents and Settings\rbove\My Documents\Minitab\2gr1-060.MTW".
Retrieving worksheet from file: 'C:\Documents and Settings\rbove\My
Documents\Minitab\2gr1-060.MTW'
Worksheet was saved on Wed Sep 13 2006
Results for: 2gr1-060.MTW
MTB > print c5
Data Display
drug3
52 53 58 50 53 58 55 66 53 51 52 53 54 55 56
MTB > Onet 'drug3';
SUBC> Confidence 99.0.
One-Sample T: drug3
Variable N Mean StDev SE Mean 99% CI
drug3 15 54.6000 3.9060 1.0085 (51.5977, 57.6023)
MTB > OneZ 'drug3';
SUBC> Sigma 4.5;
SUBC> Confidence 99.5.
One-Sample Z: drug3
The assumed standard deviation = 4.5
Variable N Mean StDev SE Mean 99.5% CI
drug3 15 54.6000 3.9060 1.1619 (51.3385, 57.8615)
MTB > Stop.
First Extra Credit Problem
4. a. Use the data above to compute a 98% confidence interval for the population standard deviation.
b. Assume that you got the sample standard deviation that you got above from a sample of 45, repeat a.
c. Fool around with the method for getting a confidence interval for a median and try to come close to a 99% confidence interval for the median.
Solution: a. Recall that and . The problem says that and . From the supplement pg 1 (or Table 3), . We use and . The formula becomes or . If we take square roots, we get .
b. We will repeat a) with . Recall . Now From the supplement pg 2 (or Table 3), the formula for large samples is . Since the table has no values for 44 degrees of freedom, we must use the large sample formula. We use and . The formula becomes or .
c. We fool around with the method for getting a confidence interval for a median and try to come close to a 99% confidence interval for the median.
The numbers in order are
50 51 52 52 53 53 53 53 54 55 55 56 58 58 66
It says on the outline that, if we use the numbers from the end, . We want to be 1% or lower which means . There are two ways to do this. If we take the easy way out and use a Normal approximation This seems to be telling us to use the numbers that are 3 from each end or 52 and 58. (To be conservative, round the result down.)
To be more precise, use the Binomial table with . Possible intervals are to , to etc. Let’s try a few intervals.
Interval
to or 50 to 66 1
to or 51 to 58 2
to or 52 to 58 3
to or 52 to 56 4
Notice that we could have answered the question by finding the largest value of with
Since the smallest interval with a significance level below 1% is 52 to 58, this is the best that we can do.
We can check our results using the Normal distribution. The outline says, using a continuity correction,
.
Since we need , was correct.
Extra Credit Minitab Problem
5. Check some numbers in the Normal, t, Chi-Squared or F tables using the new set of Minitab routines that I have prepared. To use the new set of routines, follow the instructions in Areadoc1. There are several things that you can do. For the Normal distribution use the computer to check the answers to Examples 6.1-6.4 on pp 198-200 in the text. For the t-table pick a number of degrees of freedom and show that for that number of degrees of freedom, the probability above, say, is 20%. You can do the same for the F and chi-squared tables in your book of tables. A good answer will explain what you did and contain the command dialog and graphs.
Results: I looked at the tables and found , , , , and . For the numbers with .10 as a subscript, I checked that the probability above them was .10, for the numbers with .90 as a subscript, I checked that the probability below them was .10.
————— 9/19/2005 5:33:43 PM ————————————————————
Welcome to Minitab, press F1 for help.
MTB > WOpen "C:\Documents and Settings\rbove\My Documents\Minitab\notmuch.MTW".
Retrieving worksheet from file: 'C:\Documents and Settings\rbove\My
Documents\Minitab\notmuch.MTW'
Worksheet was saved on Thu Apr 14 2005
Results for: notmuch.MTW
MTB > %tarea6a
Executing from file: tarea6a.MAC
Graphic display of t curve areas
Finds and displays areas to the left or right of a given value
or between two values. (This macro uses C100-C116 and K100-K120)
Enter the degrees of freedom.
DATA> 10
Do you want the area to the left of a value? (Y or N)
n
Do you want the area to the right of a value? (Y or N)
y
Enter the value for which you want the area to the right.
DATA> 1.372
...working...
t Curve Area
Data Display
mode 0
median 0
MTB > %normarea6a
Executing from file: normarea6a.MAC
Graphic display of normal curve areas
Finds and displays areas to the left or right of a given value
or between two values. (This macro uses C100-C116 and K100-K116)
Enter the mean and standard deviation of the normal curve.
DATA> 0
DATA> 1
Do you want the area to the left of a value? (Y or N)
n
Do you want the area to the right of a value? (Y or N)
y
Enter the value for which you want the area to the right.
DATA> 1.282
...working...
Normal Curve Area
MTB > %chiarea6a
Executing from file: chiarea6a.MAC
Graphic display of chi square curve areas
Finds and displays areas to the left or right of a given value
or between two values. (This macro uses C100-C116 and K100-K120)
Enter the degrees of freedom.
DATA> 10
Do you want the area to the left of a value? (Y or N)
n
Do you want the area to the right of a value? (Y or N)
y
Enter the value for which you want the area to the right.
DATA> 15.9872
...working...
ChiSquare Curve Area
Data Display
std_dev 4.47214
mode 8.00000
median 9.33333
MTB > %chiarea6a
Executing from file: chiarea6a.MAC
Graphic display of chi square curve areas
Finds and displays areas to the left or right of a given value
or between two values. (This macro uses C100-C116 and K100-K120)
Enter the degrees of freedom.
DATA> 10
Do you want the area to the left of a value? (Y or N)
l
Please answer Yes or No.
y
Enter the value for which you want the area to the left.
DATA> 4.8650
...working...
Chi Squared Curve Area
Data Display
std_dev 4.47214
mode 8.00000
median 9.33333
MTB > %farea6a
Executing from file: farea6a.MAC
Graphic display of F curve areas
Finds and displays areas to the left or right of a given value
or between two values. (This macro uses C100-C116 and K100-K120)
Enter the degrees of freedom.DF2 must be above 4.
DATA> 10
DATA> 10
Do you want the area to the left of a value? (Y or N)
n
Do you want the area to the right of a value? (Y or N)
y
Enter the value for which you want the area to the right.
DATA> 2.32
...working...
F Curve Area
Data Display
mode 0.818182
std dev 0.968246
MTB > %farea6a
Executing from file: farea6a.MAC
Graphic display of F curve areas
Finds and displays areas to the left or right of a given value
or between two values. (This macro uses C100-C116 and K100-K120)
Enter the degrees of freedom.DF2 must be above 4.
DATA> 10
DATA> 10
Do you want the area to the left of a value? (Y or N)
y
Enter the value for which you want the area to the left.
DATA> .431
...working...
F Curve Area
Data Display
mode 0.818182
std dev 0.968246
MTB >